Splits the existing plane_depth_sequencing paper into two:
papers/plane_depth/paper.tex (NEW, 4 pages):
- Plane depth definition.
- Level edge, up/down/neutral triangle classification.
- Outerplanarity lemma (formerly Lemma 2.6 of PDS).
- Deep embedding G' definition.
- "Every face of G' is up or down" lemma.
- Unique level edge per face; shared level edge between adjacent faces.
- Quadrilateral decomposition definition with three types
(shallow diamond, deep diamond, S quad).
papers/plane_depth_sequencing/paper.tex (slimmed from 11 → 6 pages):
- Cites plane_depth for all foundational definitions.
- Keeps: slice, move definitions (anchor drop, level add, join,
ring completion), move selection, termination theorem.
papers/coloring_nested_tire_graphs/paper.tex:
- Bibliography updated: cite bauerfeld-depth instead of bauerfeld-pds.
- Two in-text references updated to cite the new outerplanarity
lemma in plane_depth.
Rationale: the outerplanarity / deep-embedding / quadrilateral-
decomposition material is foundational and reused by multiple
papers (and by the proposed level-cycle generalization). The
quadrilateral-sequencing programme is one specific application.
Splitting lets coloring_nested_tire_graphs cite the foundations
cleanly without dragging in the sequencing machinery.
Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
In `plane_depth_sequencing/paper.tex`:
- Lemma 2.6 now allows any nonempty source S ⊆ V(G) whose vertices all
lie on the boundary of the outer face of the chosen embedding,
rather than only the outer-cycle case S = V(C).
- The proof is the same argument with S in place of C: at d=0 each
S-vertex remains on the outer face after restriction; for d ≥ 1
the BFS ball V_{<d}^S is connected and reaches the outer face
via S.
- The original outer-cycle statement is preserved as a remark inside
the lemma.
- Adds \label{lem:outerplanarity}.
In `coloring_nested_tire_graphs/paper.tex`:
- The proof of Lemma 1.7 drops the "extends verbatim" caveat and
simply cites the generalised Lemma 2.6, noting that since the level
source S is a single vertex (per the local Level-source definition)
we may freely choose an embedding placing S on the outer face;
outerplanarity is a graph property so the conclusion transfers.
Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
Extend the deep embedding to include the outer face, decompose into
quadrilaterals via level-edge pairing on the sphere, and define a
deterministic sequence built from four moves (anchor drop, level add,
join, ring completion) with a recursive lex-smallest tiebreak on the
initial quadrilateral. Attempt the termination theorem and the per-move
case analyses.
Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
didericis